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On the inverse problem of the calculus of variations in field theory. (English) Zbl 0557.70019
It has been argued by the author in two independent papers [e.g.: Ann. Phys. 140, 45-64 (1982; Zbl 0501.70020)] that theories possessing different variational descriptions form a set of ”measure zero” in the space of all theories derivable from a Lagrangian. The question arises, however, whether the field theories of physical interest do not precisely belong to that set because of their particular properties (symmetries, locality of the field equations). In this paper the beginning of an answer to that question was provided.
More precisely, given a set of second order quasilinear field equations of a fairly general class, it is proven that there exist, under suitable restrictions, a unique (up to a divergence) Lagrangian such that the corresponding variational equations are equivalent to the given set. The restrictions are that the action is an integral of a function of just the coordinates, the ”fields” and their first order partial derivatives, and that the variational principle is stated in terms of the ”fields”. The first (locality) requirement should be relaxed in further study for reasons discussed in the text.
Reviewer: W.Wreszinski

MSC:
70Sxx Classical field theories
49S05 Variational principles of physics
35R30 Inverse problems for PDEs
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